Question

A rock is thrown from one side of a river to another. The function h(t) = -16t^2 + 80t + 30 gives the height in inches of the rock t seconds after it has been thrown. What does h (2.5) represent in the context of this problem?

Answer 1

In the context of this problem, the function \( h(t) = -16t^2 + 80t + 30 \) describes the height of the rock above the ground (or the level of the river) at any given time \( t \) in seconds after it has been thrown.

When we evaluate \( h(2.5) \), we are calculating the height of the rock \( 2.5 \) seconds after it was thrown. Specifically, \( h(2.5) \) represents the height of the rock above the ground (or above the river) at that particular moment in time (2.5 seconds).

To find the actual height at that time, we would substitute \( t = 2.5 \) into the function:

\[
h(2.5) = -16(2.5)^2 + 80(2.5) + 30
\]

Calculating this gives:

\[
h(2.5) = -16(6.25) + 80(2.5) + 30
\]
\[
= -100 + 200 + 30
\]
\[
= 130
\]

Thus, \( h(2.5) = 130 \), which means the rock is 130 inches above the ground 2.5 seconds after being thrown.